"Anything placed in so-called bucket three will never pass," Cruz told reporters. "If we want to pass real reforms we have to do it now on budget reconciliation."

The White House has outlined a three-part plan for repealing and replacing ObamaCare: Using a budget process called reconciliation to repeal and replace large chunks of the law; allowing Health and Human Services Secretary Tom Price to issue regulations to bring down costs; and Congress passing a bill that would include additional reforms like selling insurance across state lines.

While Republicans are using reconciliation to gut core provisions of ObamaCare with a simple majority, they wouldn't be able to use that rule for any legislation that would fall under the so-called "third prong" of the White House plan.

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That means to pass additional healthcare reform legislation separate from the current repeal and replace bill, Senate Republicans will need the support of at least eight Senate Democrats to meet a 60-vote threshold.

But Cruz stressed that "absent a significant change" from Senate Democrats, he didn't believe that wasn't likely to happen.

"The position of Senate Democrats has been blanket obstruction," he said. "I believe we a very unlikely to get eight Senate Democrats to join in doing anything."

Democrats have repeatedly said they are willing to work with Republicans to repair, but not repeal, ObamaCare.

But Republican leadership has shown no signs of backing away from the House plan to repeal and replace ObamaCare, despite growing concerns within their own party.

Cruz has said that he doesn't believe the current House bill can pass and told reporters on Tuesday that "negotiations are ongoing" with GOP lawmakers and the Trump administration.

"My central focus is trying to improve the House bill so that it drives down the cost of premiums," he said. "I'm working actively to improve that. I'm also concerned about the current terms of Medicaid expansion."

Cruz told reporters last week that he wanted to overrule the Senate parliamentarian to expand the House plan.

House Republicans left several reforms popular with conservatives out of their healthcare bill because the parliamentarian is likely to rule them outside the scope of special rules in the upper chamber that prevent a Democratic filibuster.